src/HOL/Library/Cardinality.thy
 author haftmann Sat Oct 20 09:12:16 2012 +0200 (2012-10-20) changeset 49948 744934b818c7 parent 49689 b8a710806de9 child 51116 0dac0158b8d4 permissions -rw-r--r--
moved quite generic material from theory Enum to more appropriate places
```     1 (*  Title:      HOL/Library/Cardinality.thy
```
```     2     Author:     Brian Huffman, Andreas Lochbihler
```
```     3 *)
```
```     4
```
```     5 header {* Cardinality of types *}
```
```     6
```
```     7 theory Cardinality
```
```     8 imports Phantom_Type
```
```     9 begin
```
```    10
```
```    11 subsection {* Preliminary lemmas *}
```
```    12 (* These should be moved elsewhere *)
```
```    13
```
```    14 lemma (in type_definition) univ:
```
```    15   "UNIV = Abs ` A"
```
```    16 proof
```
```    17   show "Abs ` A \<subseteq> UNIV" by (rule subset_UNIV)
```
```    18   show "UNIV \<subseteq> Abs ` A"
```
```    19   proof
```
```    20     fix x :: 'b
```
```    21     have "x = Abs (Rep x)" by (rule Rep_inverse [symmetric])
```
```    22     moreover have "Rep x \<in> A" by (rule Rep)
```
```    23     ultimately show "x \<in> Abs ` A" by (rule image_eqI)
```
```    24   qed
```
```    25 qed
```
```    26
```
```    27 lemma (in type_definition) card: "card (UNIV :: 'b set) = card A"
```
```    28   by (simp add: univ card_image inj_on_def Abs_inject)
```
```    29
```
```    30 lemma finite_range_Some: "finite (range (Some :: 'a \<Rightarrow> 'a option)) = finite (UNIV :: 'a set)"
```
```    31 by(auto dest: finite_imageD intro: inj_Some)
```
```    32
```
```    33 lemma infinite_literal: "\<not> finite (UNIV :: String.literal set)"
```
```    34 proof -
```
```    35   have "inj STR" by(auto intro: injI)
```
```    36   thus ?thesis
```
```    37     by(auto simp add: type_definition.univ[OF type_definition_literal] infinite_UNIV_listI dest: finite_imageD)
```
```    38 qed
```
```    39
```
```    40 subsection {* Cardinalities of types *}
```
```    41
```
```    42 syntax "_type_card" :: "type => nat" ("(1CARD/(1'(_')))")
```
```    43
```
```    44 translations "CARD('t)" => "CONST card (CONST UNIV \<Colon> 't set)"
```
```    45
```
```    46 typed_print_translation (advanced) {*
```
```    47   let
```
```    48     fun card_univ_tr' ctxt _ [Const (@{const_syntax UNIV}, Type (_, [T]))] =
```
```    49       Syntax.const @{syntax_const "_type_card"} \$ Syntax_Phases.term_of_typ ctxt T
```
```    50   in [(@{const_syntax card}, card_univ_tr')] end
```
```    51 *}
```
```    52
```
```    53 lemma card_prod [simp]: "CARD('a \<times> 'b) = CARD('a) * CARD('b)"
```
```    54   unfolding UNIV_Times_UNIV [symmetric] by (simp only: card_cartesian_product)
```
```    55
```
```    56 lemma card_UNIV_sum: "CARD('a + 'b) = (if CARD('a) \<noteq> 0 \<and> CARD('b) \<noteq> 0 then CARD('a) + CARD('b) else 0)"
```
```    57 unfolding UNIV_Plus_UNIV[symmetric]
```
```    58 by(auto simp add: card_eq_0_iff card_Plus simp del: UNIV_Plus_UNIV)
```
```    59
```
```    60 lemma card_sum [simp]: "CARD('a + 'b) = CARD('a::finite) + CARD('b::finite)"
```
```    61 by(simp add: card_UNIV_sum)
```
```    62
```
```    63 lemma card_UNIV_option: "CARD('a option) = (if CARD('a) = 0 then 0 else CARD('a) + 1)"
```
```    64 proof -
```
```    65   have "(None :: 'a option) \<notin> range Some" by clarsimp
```
```    66   thus ?thesis
```
```    67     by(simp add: UNIV_option_conv card_eq_0_iff finite_range_Some card_insert_disjoint card_image)
```
```    68 qed
```
```    69
```
```    70 lemma card_option [simp]: "CARD('a option) = Suc CARD('a::finite)"
```
```    71 by(simp add: card_UNIV_option)
```
```    72
```
```    73 lemma card_UNIV_set: "CARD('a set) = (if CARD('a) = 0 then 0 else 2 ^ CARD('a))"
```
```    74 by(simp add: Pow_UNIV[symmetric] card_eq_0_iff card_Pow del: Pow_UNIV)
```
```    75
```
```    76 lemma card_set [simp]: "CARD('a set) = 2 ^ CARD('a::finite)"
```
```    77 by(simp add: card_UNIV_set)
```
```    78
```
```    79 lemma card_nat [simp]: "CARD(nat) = 0"
```
```    80   by (simp add: card_eq_0_iff)
```
```    81
```
```    82 lemma card_fun: "CARD('a \<Rightarrow> 'b) = (if CARD('a) \<noteq> 0 \<and> CARD('b) \<noteq> 0 \<or> CARD('b) = 1 then CARD('b) ^ CARD('a) else 0)"
```
```    83 proof -
```
```    84   {  assume "0 < CARD('a)" and "0 < CARD('b)"
```
```    85     hence fina: "finite (UNIV :: 'a set)" and finb: "finite (UNIV :: 'b set)"
```
```    86       by(simp_all only: card_ge_0_finite)
```
```    87     from finite_distinct_list[OF finb] obtain bs
```
```    88       where bs: "set bs = (UNIV :: 'b set)" and distb: "distinct bs" by blast
```
```    89     from finite_distinct_list[OF fina] obtain as
```
```    90       where as: "set as = (UNIV :: 'a set)" and dista: "distinct as" by blast
```
```    91     have cb: "CARD('b) = length bs"
```
```    92       unfolding bs[symmetric] distinct_card[OF distb] ..
```
```    93     have ca: "CARD('a) = length as"
```
```    94       unfolding as[symmetric] distinct_card[OF dista] ..
```
```    95     let ?xs = "map (\<lambda>ys. the o map_of (zip as ys)) (List.n_lists (length as) bs)"
```
```    96     have "UNIV = set ?xs"
```
```    97     proof(rule UNIV_eq_I)
```
```    98       fix f :: "'a \<Rightarrow> 'b"
```
```    99       from as have "f = the \<circ> map_of (zip as (map f as))"
```
```   100         by(auto simp add: map_of_zip_map)
```
```   101       thus "f \<in> set ?xs" using bs by(auto simp add: set_n_lists)
```
```   102     qed
```
```   103     moreover have "distinct ?xs" unfolding distinct_map
```
```   104     proof(intro conjI distinct_n_lists distb inj_onI)
```
```   105       fix xs ys :: "'b list"
```
```   106       assume xs: "xs \<in> set (List.n_lists (length as) bs)"
```
```   107         and ys: "ys \<in> set (List.n_lists (length as) bs)"
```
```   108         and eq: "the \<circ> map_of (zip as xs) = the \<circ> map_of (zip as ys)"
```
```   109       from xs ys have [simp]: "length xs = length as" "length ys = length as"
```
```   110         by(simp_all add: length_n_lists_elem)
```
```   111       have "map_of (zip as xs) = map_of (zip as ys)"
```
```   112       proof
```
```   113         fix x
```
```   114         from as bs have "\<exists>y. map_of (zip as xs) x = Some y" "\<exists>y. map_of (zip as ys) x = Some y"
```
```   115           by(simp_all add: map_of_zip_is_Some[symmetric])
```
```   116         with eq show "map_of (zip as xs) x = map_of (zip as ys) x"
```
```   117           by(auto dest: fun_cong[where x=x])
```
```   118       qed
```
```   119       with dista show "xs = ys" by(simp add: map_of_zip_inject)
```
```   120     qed
```
```   121     hence "card (set ?xs) = length ?xs" by(simp only: distinct_card)
```
```   122     moreover have "length ?xs = length bs ^ length as" by(simp add: length_n_lists)
```
```   123     ultimately have "CARD('a \<Rightarrow> 'b) = CARD('b) ^ CARD('a)" using cb ca by simp }
```
```   124   moreover {
```
```   125     assume cb: "CARD('b) = 1"
```
```   126     then obtain b where b: "UNIV = {b :: 'b}" by(auto simp add: card_Suc_eq)
```
```   127     have eq: "UNIV = {\<lambda>x :: 'a. b ::'b}"
```
```   128     proof(rule UNIV_eq_I)
```
```   129       fix x :: "'a \<Rightarrow> 'b"
```
```   130       { fix y
```
```   131         have "x y \<in> UNIV" ..
```
```   132         hence "x y = b" unfolding b by simp }
```
```   133       thus "x \<in> {\<lambda>x. b}" by(auto)
```
```   134     qed
```
```   135     have "CARD('a \<Rightarrow> 'b) = 1" unfolding eq by simp }
```
```   136   ultimately show ?thesis
```
```   137     by(auto simp del: One_nat_def)(auto simp add: card_eq_0_iff dest: finite_fun_UNIVD2 finite_fun_UNIVD1)
```
```   138 qed
```
```   139
```
```   140 corollary finite_UNIV_fun:
```
```   141   "finite (UNIV :: ('a \<Rightarrow> 'b) set) \<longleftrightarrow>
```
```   142    finite (UNIV :: 'a set) \<and> finite (UNIV :: 'b set) \<or> CARD('b) = 1"
```
```   143   (is "?lhs \<longleftrightarrow> ?rhs")
```
```   144 proof -
```
```   145   have "?lhs \<longleftrightarrow> CARD('a \<Rightarrow> 'b) > 0" by(simp add: card_gt_0_iff)
```
```   146   also have "\<dots> \<longleftrightarrow> CARD('a) > 0 \<and> CARD('b) > 0 \<or> CARD('b) = 1"
```
```   147     by(simp add: card_fun)
```
```   148   also have "\<dots> = ?rhs" by(simp add: card_gt_0_iff)
```
```   149   finally show ?thesis .
```
```   150 qed
```
```   151
```
```   152 lemma card_nibble: "CARD(nibble) = 16"
```
```   153 unfolding UNIV_nibble by simp
```
```   154
```
```   155 lemma card_UNIV_char: "CARD(char) = 256"
```
```   156 proof -
```
```   157   have "inj (\<lambda>(x, y). Char x y)" by(auto intro: injI)
```
```   158   thus ?thesis unfolding UNIV_char by(simp add: card_image card_nibble)
```
```   159 qed
```
```   160
```
```   161 lemma card_literal: "CARD(String.literal) = 0"
```
```   162 by(simp add: card_eq_0_iff infinite_literal)
```
```   163
```
```   164 subsection {* Classes with at least 1 and 2  *}
```
```   165
```
```   166 text {* Class finite already captures "at least 1" *}
```
```   167
```
```   168 lemma zero_less_card_finite [simp]: "0 < CARD('a::finite)"
```
```   169   unfolding neq0_conv [symmetric] by simp
```
```   170
```
```   171 lemma one_le_card_finite [simp]: "Suc 0 \<le> CARD('a::finite)"
```
```   172   by (simp add: less_Suc_eq_le [symmetric])
```
```   173
```
```   174 text {* Class for cardinality "at least 2" *}
```
```   175
```
```   176 class card2 = finite +
```
```   177   assumes two_le_card: "2 \<le> CARD('a)"
```
```   178
```
```   179 lemma one_less_card: "Suc 0 < CARD('a::card2)"
```
```   180   using two_le_card [where 'a='a] by simp
```
```   181
```
```   182 lemma one_less_int_card: "1 < int CARD('a::card2)"
```
```   183   using one_less_card [where 'a='a] by simp
```
```   184
```
```   185
```
```   186 subsection {* A type class for deciding finiteness of types *}
```
```   187
```
```   188 type_synonym 'a finite_UNIV = "('a, bool) phantom"
```
```   189
```
```   190 class finite_UNIV =
```
```   191   fixes finite_UNIV :: "('a, bool) phantom"
```
```   192   assumes finite_UNIV: "finite_UNIV = Phantom('a) (finite (UNIV :: 'a set))"
```
```   193
```
```   194 lemma finite_UNIV_code [code_unfold]:
```
```   195   "finite (UNIV :: 'a :: finite_UNIV set)
```
```   196   \<longleftrightarrow> of_phantom (finite_UNIV :: 'a finite_UNIV)"
```
```   197 by(simp add: finite_UNIV)
```
```   198
```
```   199 subsection {* A type class for computing the cardinality of types *}
```
```   200
```
```   201 definition is_list_UNIV :: "'a list \<Rightarrow> bool"
```
```   202 where [code del]: "is_list_UNIV xs = (let c = CARD('a) in if c = 0 then False else size (remdups xs) = c)"
```
```   203
```
```   204 lemma is_list_UNIV_iff: "is_list_UNIV xs \<longleftrightarrow> set xs = UNIV"
```
```   205 by(auto simp add: is_list_UNIV_def Let_def card_eq_0_iff List.card_set[symmetric]
```
```   206    dest: subst[where P="finite", OF _ finite_set] card_eq_UNIV_imp_eq_UNIV)
```
```   207
```
```   208 type_synonym 'a card_UNIV = "('a, nat) phantom"
```
```   209
```
```   210 class card_UNIV = finite_UNIV +
```
```   211   fixes card_UNIV :: "'a card_UNIV"
```
```   212   assumes card_UNIV: "card_UNIV = Phantom('a) CARD('a)"
```
```   213
```
```   214 lemma card_UNIV_code [code_unfold]:
```
```   215   "CARD('a :: card_UNIV) = of_phantom (card_UNIV :: 'a card_UNIV)"
```
```   216 by(simp add: card_UNIV)
```
```   217
```
```   218 lemma is_list_UNIV_code [code]:
```
```   219   "is_list_UNIV (xs :: 'a list) =
```
```   220   (let c = CARD('a :: card_UNIV) in if c = 0 then False else size (remdups xs) = c)"
```
```   221 by(rule is_list_UNIV_def)
```
```   222
```
```   223 subsection {* Instantiations for @{text "card_UNIV"} *}
```
```   224
```
```   225 instantiation nat :: card_UNIV begin
```
```   226 definition "finite_UNIV = Phantom(nat) False"
```
```   227 definition "card_UNIV = Phantom(nat) 0"
```
```   228 instance by intro_classes (simp_all add: finite_UNIV_nat_def card_UNIV_nat_def)
```
```   229 end
```
```   230
```
```   231 instantiation int :: card_UNIV begin
```
```   232 definition "finite_UNIV = Phantom(int) False"
```
```   233 definition "card_UNIV = Phantom(int) 0"
```
```   234 instance by intro_classes (simp_all add: card_UNIV_int_def finite_UNIV_int_def infinite_UNIV_int)
```
```   235 end
```
```   236
```
```   237 instantiation code_numeral :: card_UNIV begin
```
```   238 definition "finite_UNIV = Phantom(code_numeral) False"
```
```   239 definition "card_UNIV = Phantom(code_numeral) 0"
```
```   240 instance
```
```   241   by(intro_classes)(auto simp add: card_UNIV_code_numeral_def finite_UNIV_code_numeral_def type_definition.univ[OF type_definition_code_numeral] card_eq_0_iff dest!: finite_imageD intro: inj_onI)
```
```   242 end
```
```   243
```
```   244 instantiation list :: (type) card_UNIV begin
```
```   245 definition "finite_UNIV = Phantom('a list) False"
```
```   246 definition "card_UNIV = Phantom('a list) 0"
```
```   247 instance by intro_classes (simp_all add: card_UNIV_list_def finite_UNIV_list_def infinite_UNIV_listI)
```
```   248 end
```
```   249
```
```   250 instantiation unit :: card_UNIV begin
```
```   251 definition "finite_UNIV = Phantom(unit) True"
```
```   252 definition "card_UNIV = Phantom(unit) 1"
```
```   253 instance by intro_classes (simp_all add: card_UNIV_unit_def finite_UNIV_unit_def)
```
```   254 end
```
```   255
```
```   256 instantiation bool :: card_UNIV begin
```
```   257 definition "finite_UNIV = Phantom(bool) True"
```
```   258 definition "card_UNIV = Phantom(bool) 2"
```
```   259 instance by(intro_classes)(simp_all add: card_UNIV_bool_def finite_UNIV_bool_def)
```
```   260 end
```
```   261
```
```   262 instantiation nibble :: card_UNIV begin
```
```   263 definition "finite_UNIV = Phantom(nibble) True"
```
```   264 definition "card_UNIV = Phantom(nibble) 16"
```
```   265 instance by(intro_classes)(simp_all add: card_UNIV_nibble_def card_nibble finite_UNIV_nibble_def)
```
```   266 end
```
```   267
```
```   268 instantiation char :: card_UNIV begin
```
```   269 definition "finite_UNIV = Phantom(char) True"
```
```   270 definition "card_UNIV = Phantom(char) 256"
```
```   271 instance by intro_classes (simp_all add: card_UNIV_char_def card_UNIV_char finite_UNIV_char_def)
```
```   272 end
```
```   273
```
```   274 instantiation prod :: (finite_UNIV, finite_UNIV) finite_UNIV begin
```
```   275 definition "finite_UNIV = Phantom('a \<times> 'b)
```
```   276   (of_phantom (finite_UNIV :: 'a finite_UNIV) \<and> of_phantom (finite_UNIV :: 'b finite_UNIV))"
```
```   277 instance by intro_classes (simp add: finite_UNIV_prod_def finite_UNIV finite_prod)
```
```   278 end
```
```   279
```
```   280 instantiation prod :: (card_UNIV, card_UNIV) card_UNIV begin
```
```   281 definition "card_UNIV = Phantom('a \<times> 'b)
```
```   282   (of_phantom (card_UNIV :: 'a card_UNIV) * of_phantom (card_UNIV :: 'b card_UNIV))"
```
```   283 instance by intro_classes (simp add: card_UNIV_prod_def card_UNIV)
```
```   284 end
```
```   285
```
```   286 instantiation sum :: (finite_UNIV, finite_UNIV) finite_UNIV begin
```
```   287 definition "finite_UNIV = Phantom('a + 'b)
```
```   288   (of_phantom (finite_UNIV :: 'a finite_UNIV) \<and> of_phantom (finite_UNIV :: 'b finite_UNIV))"
```
```   289 instance
```
```   290   by intro_classes (simp add: UNIV_Plus_UNIV[symmetric] finite_UNIV_sum_def finite_UNIV del: UNIV_Plus_UNIV)
```
```   291 end
```
```   292
```
```   293 instantiation sum :: (card_UNIV, card_UNIV) card_UNIV begin
```
```   294 definition "card_UNIV = Phantom('a + 'b)
```
```   295   (let ca = of_phantom (card_UNIV :: 'a card_UNIV);
```
```   296        cb = of_phantom (card_UNIV :: 'b card_UNIV)
```
```   297    in if ca \<noteq> 0 \<and> cb \<noteq> 0 then ca + cb else 0)"
```
```   298 instance by intro_classes (auto simp add: card_UNIV_sum_def card_UNIV card_UNIV_sum)
```
```   299 end
```
```   300
```
```   301 instantiation "fun" :: (finite_UNIV, card_UNIV) finite_UNIV begin
```
```   302 definition "finite_UNIV = Phantom('a \<Rightarrow> 'b)
```
```   303   (let cb = of_phantom (card_UNIV :: 'b card_UNIV)
```
```   304    in cb = 1 \<or> of_phantom (finite_UNIV :: 'a finite_UNIV) \<and> cb \<noteq> 0)"
```
```   305 instance
```
```   306   by intro_classes (auto simp add: finite_UNIV_fun_def Let_def card_UNIV finite_UNIV finite_UNIV_fun card_gt_0_iff)
```
```   307 end
```
```   308
```
```   309 instantiation "fun" :: (card_UNIV, card_UNIV) card_UNIV begin
```
```   310 definition "card_UNIV = Phantom('a \<Rightarrow> 'b)
```
```   311   (let ca = of_phantom (card_UNIV :: 'a card_UNIV);
```
```   312        cb = of_phantom (card_UNIV :: 'b card_UNIV)
```
```   313    in if ca \<noteq> 0 \<and> cb \<noteq> 0 \<or> cb = 1 then cb ^ ca else 0)"
```
```   314 instance by intro_classes (simp add: card_UNIV_fun_def card_UNIV Let_def card_fun)
```
```   315 end
```
```   316
```
```   317 instantiation option :: (finite_UNIV) finite_UNIV begin
```
```   318 definition "finite_UNIV = Phantom('a option) (of_phantom (finite_UNIV :: 'a finite_UNIV))"
```
```   319 instance by intro_classes (simp add: finite_UNIV_option_def finite_UNIV)
```
```   320 end
```
```   321
```
```   322 instantiation option :: (card_UNIV) card_UNIV begin
```
```   323 definition "card_UNIV = Phantom('a option)
```
```   324   (let c = of_phantom (card_UNIV :: 'a card_UNIV) in if c \<noteq> 0 then Suc c else 0)"
```
```   325 instance by intro_classes (simp add: card_UNIV_option_def card_UNIV card_UNIV_option)
```
```   326 end
```
```   327
```
```   328 instantiation String.literal :: card_UNIV begin
```
```   329 definition "finite_UNIV = Phantom(String.literal) False"
```
```   330 definition "card_UNIV = Phantom(String.literal) 0"
```
```   331 instance
```
```   332   by intro_classes (simp_all add: card_UNIV_literal_def finite_UNIV_literal_def infinite_literal card_literal)
```
```   333 end
```
```   334
```
```   335 instantiation set :: (finite_UNIV) finite_UNIV begin
```
```   336 definition "finite_UNIV = Phantom('a set) (of_phantom (finite_UNIV :: 'a finite_UNIV))"
```
```   337 instance by intro_classes (simp add: finite_UNIV_set_def finite_UNIV Finite_Set.finite_set)
```
```   338 end
```
```   339
```
```   340 instantiation set :: (card_UNIV) card_UNIV begin
```
```   341 definition "card_UNIV = Phantom('a set)
```
```   342   (let c = of_phantom (card_UNIV :: 'a card_UNIV) in if c = 0 then 0 else 2 ^ c)"
```
```   343 instance by intro_classes (simp add: card_UNIV_set_def card_UNIV_set card_UNIV)
```
```   344 end
```
```   345
```
```   346 lemma UNIV_finite_1: "UNIV = set [finite_1.a\<^isub>1]"
```
```   347 by(auto intro: finite_1.exhaust)
```
```   348
```
```   349 lemma UNIV_finite_2: "UNIV = set [finite_2.a\<^isub>1, finite_2.a\<^isub>2]"
```
```   350 by(auto intro: finite_2.exhaust)
```
```   351
```
```   352 lemma UNIV_finite_3: "UNIV = set [finite_3.a\<^isub>1, finite_3.a\<^isub>2, finite_3.a\<^isub>3]"
```
```   353 by(auto intro: finite_3.exhaust)
```
```   354
```
```   355 lemma UNIV_finite_4: "UNIV = set [finite_4.a\<^isub>1, finite_4.a\<^isub>2, finite_4.a\<^isub>3, finite_4.a\<^isub>4]"
```
```   356 by(auto intro: finite_4.exhaust)
```
```   357
```
```   358 lemma UNIV_finite_5:
```
```   359   "UNIV = set [finite_5.a\<^isub>1, finite_5.a\<^isub>2, finite_5.a\<^isub>3, finite_5.a\<^isub>4, finite_5.a\<^isub>5]"
```
```   360 by(auto intro: finite_5.exhaust)
```
```   361
```
```   362 instantiation Enum.finite_1 :: card_UNIV begin
```
```   363 definition "finite_UNIV = Phantom(Enum.finite_1) True"
```
```   364 definition "card_UNIV = Phantom(Enum.finite_1) 1"
```
```   365 instance
```
```   366   by intro_classes (simp_all add: UNIV_finite_1 card_UNIV_finite_1_def finite_UNIV_finite_1_def)
```
```   367 end
```
```   368
```
```   369 instantiation Enum.finite_2 :: card_UNIV begin
```
```   370 definition "finite_UNIV = Phantom(Enum.finite_2) True"
```
```   371 definition "card_UNIV = Phantom(Enum.finite_2) 2"
```
```   372 instance
```
```   373   by intro_classes (simp_all add: UNIV_finite_2 card_UNIV_finite_2_def finite_UNIV_finite_2_def)
```
```   374 end
```
```   375
```
```   376 instantiation Enum.finite_3 :: card_UNIV begin
```
```   377 definition "finite_UNIV = Phantom(Enum.finite_3) True"
```
```   378 definition "card_UNIV = Phantom(Enum.finite_3) 3"
```
```   379 instance
```
```   380   by intro_classes (simp_all add: UNIV_finite_3 card_UNIV_finite_3_def finite_UNIV_finite_3_def)
```
```   381 end
```
```   382
```
```   383 instantiation Enum.finite_4 :: card_UNIV begin
```
```   384 definition "finite_UNIV = Phantom(Enum.finite_4) True"
```
```   385 definition "card_UNIV = Phantom(Enum.finite_4) 4"
```
```   386 instance
```
```   387   by intro_classes (simp_all add: UNIV_finite_4 card_UNIV_finite_4_def finite_UNIV_finite_4_def)
```
```   388 end
```
```   389
```
```   390 instantiation Enum.finite_5 :: card_UNIV begin
```
```   391 definition "finite_UNIV = Phantom(Enum.finite_5) True"
```
```   392 definition "card_UNIV = Phantom(Enum.finite_5) 5"
```
```   393 instance
```
```   394   by intro_classes (simp_all add: UNIV_finite_5 card_UNIV_finite_5_def finite_UNIV_finite_5_def)
```
```   395 end
```
```   396
```
```   397 subsection {* Code setup for sets *}
```
```   398
```
```   399 lemma card_Compl:
```
```   400   "finite A \<Longrightarrow> card (- A) = card (UNIV :: 'a set) - card (A :: 'a set)"
```
```   401 by (metis Compl_eq_Diff_UNIV card_Diff_subset top_greatest)
```
```   402
```
```   403 context fixes xs :: "'a :: card_UNIV list"
```
```   404 begin
```
```   405
```
```   406 definition card' :: "'a set \<Rightarrow> nat"
```
```   407 where [simp, code del, code_abbrev]: "card' = card"
```
```   408
```
```   409 lemma card'_code [code]:
```
```   410   "card' (set xs) = length (remdups xs)"
```
```   411   "card' (List.coset xs) = of_phantom (card_UNIV :: 'a card_UNIV) - length (remdups xs)"
```
```   412 by(simp_all add: List.card_set card_Compl card_UNIV)
```
```   413
```
```   414 lemma card'_UNIV [code_unfold]:
```
```   415   "card' (UNIV :: 'a :: card_UNIV set) = of_phantom (card_UNIV :: 'a card_UNIV)"
```
```   416 by(simp add: card_UNIV)
```
```   417
```
```   418 definition finite' :: "'a set \<Rightarrow> bool"
```
```   419 where [simp, code del, code_abbrev]: "finite' = finite"
```
```   420
```
```   421 lemma finite'_code [code]:
```
```   422   "finite' (set xs) \<longleftrightarrow> True"
```
```   423   "finite' (List.coset xs) \<longleftrightarrow> of_phantom (finite_UNIV :: 'a finite_UNIV)"
```
```   424 by(simp_all add: card_gt_0_iff finite_UNIV)
```
```   425
```
```   426 definition subset' :: "'a set \<Rightarrow> 'a set \<Rightarrow> bool"
```
```   427 where [simp, code del, code_abbrev]: "subset' = op \<subseteq>"
```
```   428
```
```   429 lemma subset'_code [code]:
```
```   430   "subset' A (List.coset ys) \<longleftrightarrow> (\<forall>y \<in> set ys. y \<notin> A)"
```
```   431   "subset' (set ys) B \<longleftrightarrow> (\<forall>y \<in> set ys. y \<in> B)"
```
```   432   "subset' (List.coset xs) (set ys) \<longleftrightarrow> (let n = CARD('a) in n > 0 \<and> card(set (xs @ ys)) = n)"
```
```   433 by(auto simp add: Let_def card_gt_0_iff dest: card_eq_UNIV_imp_eq_UNIV intro: arg_cong[where f=card])
```
```   434   (metis finite_compl finite_set rev_finite_subset)
```
```   435
```
```   436 definition eq_set :: "'a set \<Rightarrow> 'a set \<Rightarrow> bool"
```
```   437 where [simp, code del, code_abbrev]: "eq_set = op ="
```
```   438
```
```   439 lemma eq_set_code [code]:
```
```   440   fixes ys
```
```   441   defines "rhs \<equiv>
```
```   442   let n = CARD('a)
```
```   443   in if n = 0 then False else
```
```   444         let xs' = remdups xs; ys' = remdups ys
```
```   445         in length xs' + length ys' = n \<and> (\<forall>x \<in> set xs'. x \<notin> set ys') \<and> (\<forall>y \<in> set ys'. y \<notin> set xs')"
```
```   446   shows "eq_set (List.coset xs) (set ys) \<longleftrightarrow> rhs" (is ?thesis1)
```
```   447   and "eq_set (set ys) (List.coset xs) \<longleftrightarrow> rhs" (is ?thesis2)
```
```   448   and "eq_set (set xs) (set ys) \<longleftrightarrow> (\<forall>x \<in> set xs. x \<in> set ys) \<and> (\<forall>y \<in> set ys. y \<in> set xs)" (is ?thesis3)
```
```   449   and "eq_set (List.coset xs) (List.coset ys) \<longleftrightarrow> (\<forall>x \<in> set xs. x \<in> set ys) \<and> (\<forall>y \<in> set ys. y \<in> set xs)" (is ?thesis4)
```
```   450 proof -
```
```   451   show ?thesis1 (is "?lhs \<longleftrightarrow> ?rhs")
```
```   452   proof
```
```   453     assume ?lhs thus ?rhs
```
```   454       by(auto simp add: rhs_def Let_def List.card_set[symmetric] card_Un_Int[where A="set xs" and B="- set xs"] card_UNIV Compl_partition card_gt_0_iff dest: sym)(metis finite_compl finite_set)
```
```   455   next
```
```   456     assume ?rhs
```
```   457     moreover have "\<lbrakk> \<forall>y\<in>set xs. y \<notin> set ys; \<forall>x\<in>set ys. x \<notin> set xs \<rbrakk> \<Longrightarrow> set xs \<inter> set ys = {}" by blast
```
```   458     ultimately show ?lhs
```
```   459       by(auto simp add: rhs_def Let_def List.card_set[symmetric] card_UNIV card_gt_0_iff card_Un_Int[where A="set xs" and B="set ys"] dest: card_eq_UNIV_imp_eq_UNIV split: split_if_asm)
```
```   460   qed
```
```   461   thus ?thesis2 unfolding eq_set_def by blast
```
```   462   show ?thesis3 ?thesis4 unfolding eq_set_def List.coset_def by blast+
```
```   463 qed
```
```   464
```
```   465 end
```
```   466
```
```   467 notepad begin (* test code setup *)
```
```   468 have "List.coset [True] = set [False] \<and> List.coset [] \<subseteq> List.set [True, False] \<and> finite (List.coset [True])"
```
```   469   by eval
```
```   470 end
```
```   471
```
```   472 hide_const (open) card' finite' subset' eq_set
```
```   473
```
```   474 end
```
```   475
```